It's another word for a tackle, but it nearly always applies to a rough or even violent tackle, particularly a sliding tackle, in which the player making the tackle (usually a defender) makes contact with the player being tackled, often bringing that player down. A challenge is not necessarily a foul; it can be a perfectly legal and effective tackle. However, the word is often applied to a tackle that is a foul and may even injure the other player. A reckless or dangerous challenge, or (as here) an "awful challenge", means a bad foul, liable to cause injury and potentially resulting not just in a free kick but in the perpetrator being booked or sent off.

See here for some further details:
https://en.wikipedia.org/wiki/Sliding_tackle

--------------------------------------------------
Note added at 3 days2 hrs (2016-01-18 08:12:37 GMT) Post-grading
--------------------------------------------------

Cilian's point is fair; a challenge can be a tackle, in the sense of an attempt to win the ball from an opponent, using the legs and feet, and often is. In this particular case it is: it refers to a dangerous tackle by Gascoigne, which was a foul; he went in with his leg to win the ball but his leg was too high. This is a dangerous kind of tackle which can easily injure the opponent.

However, the word "challenge" can include things that are not tackles, such as charging or holding an opponent. These actions are not always foul play but will be if done careless, recklessly and with excessive force. So it might be more accurate to say that a "challenge" is a tackle or some other form of deliberate and more or less forceful physical contact with another player.

For further details on foul play (though without mentioning the word "challenge"), this is useful:
http://www.fifa.com/mm/document/afdeveloping/refereeing/law_...